Determinants

The determinant is a scalar value associated with a square matrix that encodes key information about the transformation it represents — whether it is invertible, how it scales volume, and its orientation.

Definition

2×2 Determinant

det [a b] = ad - bc
    [c d]

Geometrically: the signed area of the parallelogram formed by the column vectors.

3×3 Determinant (Sarrus' Rule)

det [a b c]
    [d e f] = aei + bfg + cdh - ceg - bdi - afh
    [g h i]

General Definition (Leibniz Formula)

For an n × n matrix A:

det(A) = Σ_σ sgn(σ) · a₁σ(₁) · a₂σ(₂) · ... · aₙσ(ₙ)

Sum over all permutations σ of {1, 2, ..., n}, where sgn(σ) = +1 for even permutations, -1 for odd.

This has n! terms — impractical for computation, but important theoretically.

Cofactor Expansion

The determinant can be computed by expanding along any row or column.

Cofactor Cᵢⱼ = (-1)^(i+j) · Mᵢⱼ, where Mᵢⱼ is the (i,j) minor (determinant of the submatrix obtained by deleting row i and column j).

Expansion along row i:

det(A) = Σⱼ₌₁ⁿ aᵢⱼ · Cᵢⱼ = Σⱼ₌₁ⁿ (-1)^(i+j) · aᵢⱼ · Mᵢⱼ

Expansion along column j:

det(A) = Σᵢ₌₁ⁿ aᵢⱼ · Cᵢⱼ

Choose the row or column with the most zeros to minimize computation.

Time: O(n!) naively (recursive cofactor expansion). O(n³) via Gaussian elimination.

Laplace Expansion (Generalized)

Expand along multiple rows or columns simultaneously using complementary minors. This is the Laplace expansion theorem, generalizing cofactor expansion.

Properties

Fundamental Properties

det(I) = 1
Row swap changes sign: det(..., Rᵢ, ..., Rⱼ, ...) = -det(..., Rⱼ, ..., Rᵢ, ...)
Row scaling: det(..., cRᵢ, ...) = c · det(..., Rᵢ, ...)
Row addition: det(..., Rᵢ + cRⱼ, ...) = det(..., Rᵢ, ...) (doesn't change)
Multilinearity: det is linear in each row separately

Derived Properties

det(cA) = cⁿ det(A) for n × n matrix A
det(Aᵀ) = det(A)
det(AB) = det(A) · det(B) (multiplicative)
det(A⁻¹) = 1/det(A)
det(A) ≠ 0 iff A is invertible
Triangular matrix: det = product of diagonal entries
Block triangular: det [A B; 0 D] = det(A) · det(D)

Determinant via Row Reduction

The most efficient way to compute determinants:

Row reduce to upper triangular form U.
Track row swaps (each flips sign) and row scalings.
det(A) = (-1)^(# swaps) × (product of scaling factors) × (product of diagonal of U).

Time: O(n³).

Cramer's Rule

For a system Ax = b where A is n × n and invertible:

xᵢ = det(Aᵢ) / det(A)

where Aᵢ is A with column i replaced by b.

Practical use: O(n · n!) naively, O(n⁴) with efficient det computation. Gaussian elimination (O(n³)) is faster for solving systems. Cramer's rule is mainly of theoretical importance.

Useful for: Small systems (2×2, 3×3), deriving formulas, theoretical proofs.

Adjugate Matrix

The adjugate (or classical adjoint) adj(A) is the transpose of the cofactor matrix:

adj(A)ᵢⱼ = Cⱼᵢ = (-1)^(i+j) Mⱼᵢ

Key identity:

A · adj(A) = adj(A) · A = det(A) · I

Therefore:

A⁻¹ = adj(A) / det(A)

This gives an explicit formula for the inverse but is impractical for large matrices.

Geometric Interpretation

Determinant and Volume

The absolute value |det(A)| equals the volume of the parallelepiped (or parallelogram in 2D) formed by the column vectors of A.

2D: |det([v₁ | v₂])| = area of parallelogram spanned by v₁, v₂.

3D: |det([v₁ | v₂ | v₃])| = volume of parallelepiped spanned by v₁, v₂, v₃.

nD: n-dimensional volume of the parallelotope.

Sign and Orientation

det > 0: The transformation preserves orientation (right-handed → right-handed).
det < 0: The transformation reverses orientation (reflection).
det = 0: The transformation collapses dimension (singular, not invertible).

Determinant as Scaling Factor

A linear transformation T with matrix A scales volumes by a factor of |det(A)|.

If R is a region with volume V(R), then V(T(R)) = |det(A)| · V(R).

This is the basis for the change of variables formula in multivariable calculus:

∫∫_T(R) f(u,v) du dv = ∫∫_R f(T(x,y)) |det(J)| dx dy

where J is the Jacobian matrix.

Special Determinant Formulas

Vandermonde determinant:

det [1   1   ...  1  ]
    [x₁  x₂  ... xₙ ]  = Π_{i<j} (xⱼ - xᵢ)
    [x₁² x₂² ... xₙ²]
    [ ⋮    ⋮   ⋱   ⋮ ]
    [x₁ⁿ⁻¹ ... xₙⁿ⁻¹]

Non-zero iff all xᵢ are distinct. Appears in polynomial interpolation.

Circulant determinant: For a circulant matrix C with first row (c₀, c₁, ..., cₙ₋₁):

det(C) = Π_{k=0}^{n-1} p(ωᵏ)

where p(x) = c₀ + c₁x + ... + cₙ₋₁xⁿ⁻¹ and ω = e^(2πi/n).

Determinant Identities

Matrix determinant lemma: For invertible A ∈ ℝⁿˣⁿ, vectors u, v ∈ ℝⁿ:

det(A + uvᵀ) = (1 + vᵀA⁻¹u) · det(A)

Rank-1 update of the determinant in O(n²) instead of O(n³).

Sylvester's determinant identity: For A ∈ ℝᵐˣⁿ, B ∈ ℝⁿˣᵐ:

det(Iₘ + AB) = det(Iₙ + BA)

Useful when m ≠ n — compute the determinant in the smaller dimension.

Real-World Applications

Invertibility testing: det(A) ≠ 0 iff the system has a unique solution.
Volume computation: Determinants compute areas, volumes, and higher-dimensional content.
Cross product: In 3D, u × v = det [e₁ e₂ e₃; u₁ u₂ u₃; v₁ v₂ v₃] (symbolic expansion).
Eigenvalue computation: Eigenvalues are roots of det(A - λI) = 0.
Jacobian in optimization: det(J) measures local volume distortion. Critical for change of variables in integration and probability.
Linear independence: det ≠ 0 iff columns are linearly independent.
Orientation testing: In computational geometry, the sign of a 3×3 determinant determines if three points make a left or right turn.
Kirchhoff's theorem: The number of spanning trees of a graph = any cofactor of the Laplacian matrix (a determinant computation).